PChem Teaching Lab | Group Theory

D_4h E 2C₄ (z) C₂ 2C'₂ 2C''₂ i 2S₄ _h 2_v 2_d Linear Functions,
Rotations Quadratic
Functions Cubic
Functions

A_1g +1 +1 +1 +1 +1 +1 +1 +1 +1 +1 - x²+y², z² -

A_2g +1 +1 +1 -1 -1 +1 +1 +1 -1 -1 R_z - -

B_1g +1 -1 +1 +1 -1 +1 -1 +1 +1 -1 - x²-y² -

B_2g +1 -1 +1 -1 +1 +1 -1 +1 -1 +1 - xy -

E_g +2 0 -2 0 0 +2 0 -2 0 0 (R_x, R_y) (xz, yz) -

A_1u +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 - - -

A_2u +1 +1 +1 -1 -1 -1 -1 -1 +1 +1 z - z³, z(x²+y²)

B_1u +1 -1 +1 +1 -1 -1 +1 -1 -1 +1 - - xyz

B_2u +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 - - z(x²-y²)

E_u +2 0 -2 0 0 -2 0 +2 0 0 (x, y) - (xz², yz²) (xy², x²y), (x³, y³)

D_4h	E	2C₄ (z)	C₂	2C'₂	2C''₂	i	2S₄	_h	2_v	2_d	Linear Functions, Rotations	Quadratic Functions	Cubic Functions
A_1g	+1	+1	+1	+1	+1	+1	+1	+1	+1	+1	-	x²+y², z²	-
A_2g	+1	+1	+1	-1	-1	+1	+1	+1	-1	-1	R_z	-	-
B_1g	+1	-1	+1	+1	-1	+1	-1	+1	+1	-1	-	x²-y²	-
B_2g	+1	-1	+1	-1	+1	+1	-1	+1	-1	+1	-	xy	-
E_g	+2	0	-2	0	0	+2	0	-2	0	0	(R_x, R_y)	(xz, yz)	-
A_1u	+1	+1	+1	+1	+1	-1	-1	-1	-1	-1	-	-	-
A_2u	+1	+1	+1	-1	-1	-1	-1	-1	+1	+1	z	-	z³, z(x²+y²)
B_1u	+1	-1	+1	+1	-1	-1	+1	-1	-1	+1	-	-	xyz
B_2u	+1	-1	+1	-1	+1	-1	+1	-1	+1	-1	-	-	z(x²-y²)
E_u	+2	0	-2	0	0	-2	0	+2	0	0	(x, y)	-	(xz², yz²) (xy², x²y), (x³, y³)

Consider each symmetry element in turn, and answer how many unchanged bonds there are in each case after the symmetry element has been completed. This example is quite tricky but it will help to make a model, or to use the visualiser on the Otterbein Website.

The first number indicates the number of unchanged bonds for E, the second number
indicates the number of unchanged bonds for the C₄ rotation and so forth.

WEB TUTORIAL - Group Theory